Asymptotically optimal robust information-based quick detection for general stochastic models with nonparametric postchange uncertainty

Abstract

Abstract By making use of Kullback-Leibler information, we develop a new approach for the quickest detection problem for general statistical models with dependent observations and unknown postchange distributions; the postchange distribution depends on either unknown informative parameters or unknown nonparametric infinite-dimensional nuisance functions. For such models, we introduce a robust risk as the supremum of the mean detection delay over the class of postchange distributions. On the basis of the window-limited cumulative sum rules developed by Lai in 1988, we propose new detection procedures, making use of the noise density that minimizes the Kullback-Leibler divergence. Then for the constructed detection procedures, we provide sufficient conditions on the considered statistical models that ensure minimax optimality properties with respect to the robust risk. We apply the developed methods to the quick detection problems for both scalar and multivariate autoregressive processes with unknown postchange parameters and unknown noise distributions.